Quantum Field Theory I

2.2. CANONICAL QUANTIZATION 67
Quasilocalized states
So far, the quantum field have played a role of merely an auxiliary quantity,
appearingin the processofthe canonicalquantization. The creationand annihilation
operators look more ”physical”, since they create or annihilate physically
well defined states (of course, only to the extent to which we consider the states
with the sharp momentum being well defined). Nevertheless the fields will appear
again and again, so after a while one becomes so accustomed to them, that
one tends to consider them to be quite natural objects. Here we want to stress
that besides this psychological reason there is also a good physical reason why
the quantum fields really deserve to be considered ”physical”.
Let us consider the Fourier transform of the creation operator
∫
a + d 3 ∫
p
(x) =
(2π) 3e−i⃗p.⃗x a + ⃗p (t) = d 3 p
(2π) 3eipx a + ⃗p
Actingonthevacuumstateoneobtainsa + (x)|0〉 = ∫ d 3 p
e ipx √ 1 |⃗p〉, which
(2π) 3 2ω−⃗p
is thesuperposition(ofnormalizedmomentumeigenstates)correspondingtothe
state of the particle localized at the point x. This would be a nice object to
deal with, was there not for the unpleasant fact that it is not covariant. The
state localized at the point x is in general not Lorentz transformed 21 to the
state localized at the point Λx. Indeed
∫
a + (x)|0〉 →
d 3 p 1
(2π) 3 √ e ipx |Λp〉
2ω⃗p
andthis isingeneralnotequaltoa + (Λx)|0〉. Theproblemisthe non-invariance
of d 3 p/ √ 2ω ⃗p . Were it invariant, the substitution p → Λ −1 p would do the job.
Now let us consider ϕ(x)|0〉 = ∫ d 3 p 1
(2π) 3 2E p
e ipx |p〉 where E p = ω ⃗p = p 0
∫
ϕ(x)|0〉 →
∫
=
d 3 ∫
p 1
(2π) 3 e ipx |Λp〉 p→Λ−1 p d 3 Λ −1 p 1
=
2E p (2π) 3 e ∣ i(Λ−1 p)x ΛΛ −1 p 〉
2E Λ −1 p
d 3 p
(2π) 3 1
2E p
e i(Λ−1 p)(Λ −1 Λx) |p〉 =
∫
d 3 p 1
(2π) 3 e ip(Λx) |p〉 = ϕ(Λx)|0〉
2E p
so this object is covariant in a well defined sense. On the other hand, the state
ϕ(x)|0〉 is well localized, since
∫
〈0|a(x ′ d 3 p 1
)ϕ(x)|0〉 = 〈0|
(2π) 3 √ e ip(x−x′) |0〉
2ω⃗p
and this integral decreases rapidly for |⃗x−⃗x ′ | greater than the Compton wavelength
of the particle /mc, i.e.1/m. (Exercise: convince yourself about this.
Hint: use Mathematica or something similar.)
Conclusion: ϕ(x)|0〉 is a reasonable relativistic generalization of a state of
a localized particle. Together with the rest of the world, we will treat ϕ(x)|0〉
as a handy compromise between covariance and localizability.
21 We are trying to avoid the natural notation a + (x)|0〉 = |x〉 here, since the symbol |x〉 is
reserved for a different quantity in Peskin-Schroeder. Anyway, we want to use it at least in
this footnote, to stress that in this notation |x〉 ↛ |Λx〉 in spite of what intuition may suggest.
This fact emphasizes a need for proof of the transformation |p〉 ↛ |Λp〉 which is intuitively
equally ”clear”.